Question

AB=AD ANGLE1=ANGLE2 AND ANGLE 3=ANGLE 4 TO PROVE AP=AQ

Answer 1

By using congrencie rule you can prove them congrent and by cpct AP=AQ

Answer 2

$\huge\underline\mathfrak{Solution:}$

We have

$\angle$1=$\angle$2

$\angle$3=$\angle$4

$\implies$$\angle$1+$\angle$3=$\angle$2+$\angle$4

$\implies$$\angle$BAC=$\angle$DAC

In ∆ABC and ∆ADC, we have

AB = AD

$\angle$BAC=$\angle$DAC

AC = AC

So, by SAS-rule of congruence

∆ABC congruent ∆ADC

$\implies$BC = DC

and $\angle$ABC = $\angle$DAC

$\implies$ABP = $\angle$ADQ

In ∆ABP and ∆ADQ, we have

AB = AD

$\angle$1=$\angle$2

and $\angle$ABP = $\angle$ADQ

So, by ASA-rule of congruence

∆ABP congruent ∆ADQ

$\implies$AP = AQ